Measurement Model Scaling and Identification

Conventional Drawing and Labeling of SEM Model

SEM nomenclature conventions

name	upper case	lower case	usage
Lambda	$\Lambda$	$\lambda$	Loading of a manifest indicator onto a latent construct
Psi	$\Psi$	$\psi$	residual variance/covariance of contruct when endogenous
Theta	$\Theta$	$\theta$	residual variance/covariance of indicators
Sigma	$\Sigma$	$\sigma$	$\Sigma$ is the model implied variance/covariance matrix;
			$\sigma$ is standard deviation, $\sigma^2$ variance of indicator.
			$\sigma$ can also be covariance of indicator

\[ \textbf{$\Sigma$} = \left[ \begin{array}{cccccc} \sigma_{1,1}^2 & \sigma_{1,2} & \sigma_{1,3} & \sigma_{1,4} & \sigma_{1,5} & \sigma_{1,6} \\ \sigma_{2,1} & \sigma_{2,2}^2 & \sigma_{2,3} & \sigma_{2,4} & \sigma_{2,5} & \sigma_{2,6} \\ \sigma_{3,1} & \sigma_{3,2} & \sigma_{3,3}^2 & \sigma_{3,4} & \sigma_{3,5} & \sigma_{3,6} \\ \sigma_{4,1} & \sigma_{4,2} & \sigma_{4,3} & \sigma_{4,4}^2 & \sigma_{4,5} & \sigma_{4,6} \\ \sigma_{5,1} & \sigma_{5,2} & \sigma_{5,3} & \sigma_{5,4} & \sigma_{5,5}^2 & \sigma_{5,6} \\ \sigma_{6,1} & \sigma_{6,2} & \sigma_{6,3} & \sigma_{6,4} & \sigma_{6,5} & \sigma_{6,6}^2 \end{array} \right],\]

\[ \textbf{$\Lambda$} = \left[ \begin{array}{cc} \lambda_{1,1} & \lambda_{1,2} \\ \lambda_{2,1} & \lambda_{2,2} \\ \lambda_{3,1} & \lambda_{3,2} \end{array} \right],\]

\[ \textbf{$\Psi$} = \left[ \begin{array}{cc} \psi_{1,1} & \psi_{1,2} \\ \psi_{2,1} & \psi_{2,2} \end{array} \right],\]

\[ \textbf{$\Lambda^\prime$} = \left[ \begin{array}{cc} \lambda_{1,1} & \lambda_{2,1} & \lambda_{3,1} \\ \lambda_{1,2} & \lambda_{2,2} & \lambda_{3,2} \end{array} \right],\]

\[ \textbf{$\Theta$} = \left[ \begin{array}{cccccc} \theta_{1,1} & 0 & 0 & 0 & 0 & 0 \\ 0 & \theta_{2,2} & 0 & 0 & 0 & 0 \\ 0 & 0 & \theta_{3,3} & 0 & 0 & 0 \\ 0 & 0 & 0 & \theta_{4,4} & 0 & 0 \\ 0 & 0 & 0 & 0 & \theta_{5,5} & 0 \\ 0 & 0 & 0 & 0 & 0 & \theta_{6,6} \end{array} \right].\]

Fundamental SEM equation

\[ \Sigma = \Lambda \Psi \Lambda' + \Theta \tag{1} \]

Unidimensional Measurement Model

library(lavaan)

##Prepare data with sufficient statisitics##
mymeans<-matrix(c(3.06893, 2.92590, 3.11013), ncol=3,nrow=1)
mysd<-c(0.84194,0.88934,0.83470)
mat <- c(1.00000,
         0.55226, 1.00000,
         0.56256, 0.66307, 1.00000)
mycor <- getCov(mat, lower = TRUE)

##Transform correlation matrix to covariance matrix using information above##
mycov <- mysd %*% t(mysd)

rownames(mycor) <-c( "Glad", "Cheerful", "Happy")
colnames(mycor) <-c( "Glad", "Cheerful", "Happy")

rownames(mycov) <-c( "Glad", "Cheerful", "Happy")
colnames(mycov) <-c( "Glad", "Cheerful", "Happy")
mynob<-823

Descriptive Statistics
	Glad	Cheerful	Happy
Glad	1.00	0.55	0.56
Cheerful	0.55	1.00	0.66
Happy	0.56	0.66	1.00
Mean	3.07	2.93	3.11
SD	0.84	0.89	0.83
Var	0.71	0.79	0.70

Measurement Model: 3 indicators

using correlations only (instead of variance/covariance matirx)

Found more than one class "Model" in cache; using the first, from namespace 'MatrixModels'

lavaan (0.5-20) converged normally after   9 iterations

  Number of observations                           823

  Estimator                                         ML
  Minimum Function Test Statistic                0.000
  Degrees of freedom                                 0

Model test baseline model:

  Minimum Function Test Statistic                0.000
  Degrees of freedom                                 0
  P-value                                           NA

User model versus baseline model:

  Comparative Fit Index (CFI)                    1.000
  Tucker-Lewis Index (TLI)                       1.000

Loglikelihood and Information Criteria:

  Loglikelihood user model (H0)              -1070.768
  Loglikelihood unrestricted model (H1)      -1070.768

  Number of free parameters                          1
  Akaike (AIC)                                2143.536
  Bayesian (BIC)                              2148.249
  Sample-size adjusted Bayesian (BIC)         2145.074

Root Mean Square Error of Approximation:

  RMSEA                                          0.000
  90 Percent Confidence Interval          0.000  0.000
  P-value RMSEA <= 0.05                          1.000

Standardized Root Mean Square Residual:

  SRMR                                           0.000

Parameter Estimates:

  Information                                 Expected
  Standard Errors                             Standard

Latent Variables:
                   Estimate  Std.Err  Z-value  P(>|z|)
  Positive =~                                         
    Cheerful          1.000                           

Variances:
                   Estimate  Std.Err  Z-value  P(>|z|)
    Positive          0.790    0.039   20.285    0.000
    Cheerful          0.000

cat(file = 'topics/2_MeasurementModel/2b_ConfirmatoryFactorAnalysis/mplus/l.cheer.out')

Warning in lav_samplestats_from_moments(sample.cov = sample.cov,
sample.mean = sample.mean, : lavaan WARNING: sample covariance can not be
inverted

lavaan (0.5-20) converged normally after  99 iterations

  Number of observations                           823

  Estimator                                         ML
  Minimum Function Test Statistic                0.000
  Degrees of freedom                                 0

Parameter Estimates:

  Information                                 Expected
  Standard Errors                             Standard

Latent Variables:
                   Estimate  Std.Err  Z-value  P(>|z|)
  Positive =~                                         
    Glad              0.841    0.021   40.570    0.000
    Cheerful          0.889    0.022   40.570    0.000
    Happy             0.834    0.021   40.570    0.000

Variances:
                   Estimate  Std.Err  Z-value  P(>|z|)
    Glad              0.000    0.000   13.060    0.000
    Cheerful          0.000    0.000   12.213    0.000
    Happy             0.000    0.000   13.191    0.000
    Positive          1.000